Re: [HoTT] Identity versus equality

[Nicolai Kraus <nicola...@gmail.com>]

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On Sun, May 10, 2020 at 4:35 PM Ulrik Buchholtz <ulrikbu...@gmail.com>
wrote:

> No need to apologize: I know I was being slightly provocative by
> juxtaposing a question about sets cover (SC) and a comment on 2-level type
> theory in this context, in order to provoke some discussion :-)
>

It worked :-)


> Wouldn't you agree, however, that even though basic 2LTT is conservative
> over HoTT, from a certain point of view, 2LTT privileges the “ground floor”
> exosets? In your very nice paper, https://arxiv.org/abs/1705.03307, you
> decorate the inner (fibrant, endo-) types as special, and leave the
> exotypes undecorated, privileging the latter. While from a user's
> perspective, however, it's the (inner) types that are
> standard/mathematical, and the exotypes that are special.
>

I think I see where you are coming from. But for me, decorating the inner
types as special was simply a pragmatic choice, not a philosophical one.
Since most/all statements in the paper are "proper" 2LTT, there are more
exo- / outer types involved than endo- / inner ones. But as a user, one is
interested in the fibrant types (and maybe even assumes that they
coincide with the inner theory), and only adds some small exo-sprinkles
like "exo-Nat is cofibrant"; then, it makes sense to decorate the exo-types
instead, as e.g. in https://arxiv.org/abs/2004.06572
And maybe it would be less confusing if we did the same in the paper that
you linked to. I'm not sure.


> And regardless of the decorations, the mere fact that we bring in the
> exoset level makes the theory harder to justify from the philosophical
> position that general homotopy types are not reducible to sets. One can in
> fact see the conservativity result as foundational reduction (in the sense
> of https://math.stanford.edu/~feferman/papers/reductive.pdf section 5)
> from a system justified by the principle that everything is based on sets
> to a system justified by a framework where that isn't the case.
>

That's interesting, thanks for the link!


> Another connection is that it seems it should be easier to find an axiom,
> which might imply SC, that would allow us to construct the type of
> semi-simplicial types, rather than such an axiom that doesn't imply SC. But
> I don't know any such axiom statable in book HoTT either way.
>

Good question.


> BTW, you probably meant “simplicial set” above. And yes, that kind of
> axiom would be the strongest expression of “everything is based sets”, and
> it currently needs 2LTT to even be stated.
>

You're right, I meant "set". (Otherwise it'd be silly, a type X is the
realization of the [fibrant replacement of] the constant presheaf X.)

Nicolai


> Cheers,
> Ulrik
>
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